business maths linear programming

Question

A company manufactures two types of window frames: Timber frames and Composite
frame. The time available for some of the key manufacturing stages is limited. At
present only 2000 hours of cutting time and 1000 hours of assembly time are available
each month. The following table shows the time required to produce each frame in two
manufacturing stages.

 

 

Manufacturing Stage                Time (in hours)
                                   Timber frame Composite frame
Cutting                                     5              2
Assembly                                 2              2

 

 

The profits for each Timber frame and Composite frame are RM 400 and RM 100
respectively.

(i) Formulate a linear programming by stating the objective function and all the
constraints.

(ii) Solve the linear programming graphically to obtain the optimal product mix and
the maximum monthly profit contribution.

(iii) Suppose that the labour force in the cutting department is prepared to work
overtime, would it profitable for the company to work overtime if an overtime
premium of RM 70 an hour is paid?

(iv) If to work overtime in the cutting department, how many extra hours would you
recommend and what would be the extra profit from these hours?

Answer 1

Let the timber frame be T and the composite frame be C.

The objective function is

i) 

400 T + 100 C (profits for the timber and composite frame)

The constraints are:

5T + 2C <= 2000 (cutting constraint)

2T + 2C <= 1000 (assembly constraint)

T, C >= 0

ii)

Graph the lines above

The first line has intercepts of (1000, 0) and (0, 400)

The second line has intercepts of (500, 0) and (0, 500)

Solving

5T + 2C = 2000

2T + 2C = 1000

we subtract, getting 3T = 1000; T = 1000/3 = 333 1/3; As T + C = 500, C = 500/3 = 166 2/3

Intersection at (166 2/3, 333 1/3)

 

At (0, 400) 100C + 400T = 160000

 

At (166 2/3, 333 1/3) 100C + 400T = 150000

At (500, 0) 100C + 400T = 50000

 

You can solve it graphically by seeing the isoprofit line has slope -1/4 and thus is furthest out at the point (0, 400)

Plot 100C + 400T = 160000 to show that this line is furthest out.

Note that the cutting constraint is the active constraint.

iii)  Since it requires 5 hours of labor for cutting the timber, this will cost 70*5 = 350 extra.  The profit is 400, and 400 > 350.  Thus, it is profitable to work overtime.

iv)  There is also an assembly constraint.  Based upon this, we can only make 500 timber frames.  Thus, we want to increase our number of cutting hours by 5 * (500 - 400) = 5 * 100 = 500 hours.

The extra profit is 100 * 50 = 5000